Self-excited oscillation under nonlinear feedback with time-delay

Several important applications use nonlinear feedback methods for synthetically inducing self-excited oscillations in mechanical systems. The van der Pol and saturation function type feedback methods are widely used. The effects of time-delay on the self-excited oscillation of single and two degrees-of-freedom systems under nonlinear feedback have been studied in this paper. It is shown that a single degree-of-freedom oscillator with the van der Pol type nonlinear feedback can produce unbounded response in presence of time-delay. In general, an uncontrolled time-delay in the feedback changes the state of oscillations in an uncertain manner. Therefore, a bounded saturation type feedback with controllable time-delay is proposed for inducing self-excited oscillations. The feedback signal is essentially an infinite weighted sum of a nonlinear function of the state variables of the system measured at equal intervals in the past. More recent is the measurement, higher is the weight. Thus, the feedback signal uses a large amount of information about the past history of the dynamics. Such a control signal can be realized in practice by a recursive means. The control law allows three parameters to be varied namely, the time-delay, feedback and recursive gains. Multiple time scale analysis is used to plot amplitude vs. time-delay curves. Time-delay can be controlled to vary the amplitude of oscillation as well as to switch the oscillation from one mode to the other in a two degrees-of-freedom system. It is shown that a higher recursive gain can exercise a better and a more robust control on the amplitude of oscillation of the system. Analytical results are compared with the results of numerical simulations.